For usual graphs on $n$ vertices, a edge-minimal connected graph is nothing but a spanning tree of this graph. It is well-known that any spanning tree has $n-1$ edges.

I would like to know whether there is a similar definition of connectivity for $3$-uniform hypergraphs which preserves this property. To be more specific, is there any known definition of connectivity for $3$-uniform hypergraphs such that **every edge-minimal connected graph on $n$ vertices has exactly $\binom{n-1}{2}$ edges**?

I personally have a definition for any $k$-uniform hypergraphs which satisfies this property, and I just want to know whether some definitions have already been posed before. Thanks!